Probability in Length, Area, and Volume

It is given that there are $15$ yellow squares of $2$ square feet each and $18$ gray squares of $3$ square feet each. Therefore, by calculating the product of $15$ and $2,$ as well as $18$ and $3,$ the total areas of yellow and gray regions can be found. The sum of these values is the total area of the kitchen floor. Note that the whole area of the kitchen floor is the sample space of all the possible outcomes where the candy can fall. Since the probability of a candy falling on gray region should be calculated, the area of success region is the area of gray region. Therefore, the area of success region can be substituted with $54$ and the number of total region can be substituted with $84$ into the Probability Formula. The probability of the candy falling on the gray area of the floor is approximately $0.64$ or $64\%.$

First, the total areas of the forests, fields, and lakes should be calculated by multiplying the area of each geographical feature by the number of those features.

Object	Number	Area	Total Area
Forests	$8$	$1200$	$\stackrel{8\cdot 1200=}{9600}$
Fields	$24$	$750$	$\stackrel{24\cdot 750=}{18000}$
Lakes	$76$	$3$	$\stackrel{76\cdot 3=}{228}$

It is given that the total area of the country is $30000$ square kilometers. Since it is also given that Magdalena is not in a lake, the area of total region she could be in is the difference between the total area of the country and the total area of lakes. Magdalena must be somewhere in a $29772$ square kilometer area. To calculate the probability of the event of her being lost in a forest, substitute $9600,$ the total area of land covered by forests, for the area of success region and $29772$ for the area of total region. The probability of Magdalena being lost in a field can be calculated similarly by substituting $18000,$ the total land area made up of fields, for the area of success region and the same number for the area of total region. Therefore, the probability of Magdalena being lost in a forest is approximately $0.32,$ or $32\%,$ while the probability of her being lost in a field is about $0.60,$ or $60\%.$

The geometric probability formula can be used to find the total length of the fence. Since the probability of the event of turning and successfully finding a place in the grass is given, the length of success region is the length of parts without the fence. By substituting the length of total region with $170$ and $P$ with $\frac{3}{10},$ this value can be calculated. It can be concluded that the length of the parts without the fence is $51$ feet. The parts of the path without the fence form the sample space of the event of turning successfully. By subtracting $51$ from the length of the path, the total length of the fence can be found. Therefore, the length of the fence is $119$ feet.

The experiment uses a sphere, a three-dimensional object, so the geometric probability formula for volume should be used. The sample space in this case is the whole region inside the sphere. The volume of total region is the same as the volume of the sphere. Let $r$ be the radius of the sphere. Its volume is then determined by the following formula. Outcomes in which the bubble is situated closer to the outside than the center of the sphere are said to be the success outcomes. They can be calculated as the difference between the total volume of the sphere and the outcomes in which the bubble is situated closer to the center of the sphere. The bubble is closer to the center if it is in the sphere whose radius is half the radius of the whole sphere. The radius of this smaller sphere is $\frac{r}{2}.$ This means that the volume of that sphere is as follows. Now the volume of the success outcomes can be found. Finally, the found volumes can be substituted into the probability formula. It can be concluded that the probability of the bubble being closer to the outside of the sphere is $\frac{7}{8}.$

First, draw a graph that represents the possible times in which Magdalena and Tiffaniqua could meet. If Tiffaniqua arrives at $14\text{:}30$ $\text{P.M.},$ then Magdalena must arrive no later than $14\text{:}45$ $\text{P.M.}.$ If Tiffaniqua arrives at $14\text{:}31$ $\text{P.M.},$ Magdalena must arrive no later than $14\text{:}46$ $\text{P.M.},$ and so on. Therefore, the $15\text{-}$minute leeway shown on the graph represents their opportunity of meeting. The geometric probability formula for area will be used to calculate the probability of both girls being in that $15\text{-}$minute span.

Area of the Total Region

The total time available for the girls to meet or a sample space is represented by the square shown in the diagram. Its sides represent the $90\text{-}$minute time span from $14\text{:}30$ $\text{P.M.}$ to $16\text{:}00$ $\text{P.M.}.$ The area of the total region can be calculated by substituting $90$ for $s$ into the area of a square formula.

Area of the Success Region

Analyzing the diagram, it can be noted that the area of the $15\text{-}$minute time allowance is equal to the difference between the area of the square and the areas of the top and bottom triangles. Also, the areas of the top and bottom triangles are the same, so calculating only one of them would be enough. As can be seen, these are right triangles whose legs represent $75$ minutes. By using the area of a triangle formula, their areas can be calculated. Now, the area of the $15\text{-}$minute leeway zone can be found. Therefore, the area of success region is $2475.$

Calculating Probability

Finally, by substituting $2475$ for the area of the success region and $8100$ for the area of the total region, the probability of the event of the girls meeting can be calculated. It can be concluded that the probability that Magdalena and Tiffaniqua will meet at the library is $\frac{11}{36}.$

To use the formula, first the areas of the success region — the red region — and total regions should be found. For the purposes of the solution the circles on the board can be named. First, the radii of all the circles can be found. It is given that the diameter of the small red circle $C_1$ at the center of the dartboard is $2$ centimeters, so its radius is $\frac{2}{2}=1$ centimeter. The diameter of the green circle $C_2$ is said to be $2$ centimeters greater than the diameter of $C_1$. This implies that the diameter of $C_2$ is $2+2=4$ centimeters and its radius is $2$ centimeters. It is given that the length between $C_2$ and the next ring is $8$ centimeters, so the radius of $C_3$ is obtained by adding $2$ and $8.$ Next, by adding the width of the rectangles to the radius of $C_3,$ it can be concluded that the radius of $C_4$ is $11.5$ centimeters. The radius of $C_5$ is equal to the sum of $14$ and radius of $C_2,$ which is $2$ centimeters. Finally, the radius of $C_6$ is $1.5$ centimeters greater than $r_{C_5}.$ Now that all the radii are known, they can be substituted into the circle area formula. First, the area of $C_1$ can be found. Leave the areas in terms of pi to get a precise result at the end. Similarly, the areas of the rest of the circles can be calculated.

Circle	Radius	$\pi r^2$	Area
$C_1$	$1$	$\pi (1{)}^2$	$\pi$
$C_2$	$2$	$\pi (2{)}^2$	$4\pi$
$C_3$	$10$	$\pi (10{)}^2$	$100\pi$
$C_4$	$11.5$	$\pi (11.5{)}^2$	$132.25\pi$
$C_5$	$16$	$\pi (16{)}^2$	$256\pi$
$C_6$	$17.5$	$\pi (17.5{)}^2$	$306.25\pi$

By calculating the difference between the areas of $C_4$ and $C_3,$ the total area of smaller red and green rectangles can be found. Since there are $20$ rectangles and they all have the same size, the area of each rectangle can be calculated by dividing $23.25\pi {\text{ cm}}^2$ by $20.$ $10$ of the $20$ rectangles are red, so the total area of the red rectangles is $16.125\pi$ square centimeters. Finally, the total area of the greater green and red rectangles can be found as the difference between the areas of $C_6$ and $C_5.$ The area of each bigger rectangle can be obtained by dividing this value by $20.$ Just as in case with smaller rectangles, there are $10$ bigger red rectangles, so the total area of the bigger red rectangles is $25.125$ square centimeters. The total red region on the board consists of $1$ $\text{small}$ $\text{circle},$ $10$ $\text{small}$ $\text{rectangles}$ and $10$ $\text{big}$ $\text{rectangles}.$ Therefore, the total area of the red region is the sum of these areas. The area of the success region was found to be $42.25$ square centimeters. The area of the total region is the area of the largest circle $C_6.$ With this information in mind, the probability of the dart hitting a red region can be found. The probability of the dart hitting a red region is about $0.138,$ or $13.8\%.$ The total area of the success region is $86.5\pi$ square centimeters. The area of total region is the same as before, $306.25\pi$ square centimeters. Now enough information has been calculated to find the probability of hitting a red or green region. The probability of hitting a red or green region is approximately $0.282,$ or $28.2\%.$ The probability of hitting a black or white region is approximately $0.718,$ or $71.8\%.$

It is given that there are $30$ people in the pool, each occupying $2$ square meters. By multiplying these values, the area occupied by all the people in the pool can be calculated. The area of the pool occupied by other swimmers is $60$ square meters. The Probability Formula can be used to calculate the probability of Magdalena bumping into another person. The area of the success region and the area of the total region should be substituted with $60$ and $150,$ respectively. It can be concluded that the probability of Magdalena bumping into a person in the pool is $\frac{2}{5}.$